5 Hydrogen Atoms Interaction with Graphene: Background
5.1 Introduction to graphene properties related to atom scattering
Graphene is a single two-dimensional layer of carbon atoms bound in a hexagonal lattice structure. It is a basic building block of graphitic materials of all other dimensionalities. It can be wrapped into 0D fullerenes, rolled into 1D nanotube or stacked into 3D graphite. Graphene studies have a history of nearly 70 years. As early as 1947, Wallace worked out the electronic structure of graphene (71). A long time after that, graphene was theoretically researched as an
‘academic’ material. Since 1970, graphene layers formed on transition metal surfaces have been
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studied using low energy electron diffraction (LEED) (72), electron energy loss spectroscopy (EELS) (73) and scanning microscopies (74). But at that time, graphene was mostly an unwanted impurity in surface science research. The first isolation of a graphene monolayer was in 2004 and led to the Nobel Prize in 2010 (75).
Lattice structure. Figure 5.1 (a) shows the Bravais lattice structure of a graphene layer (76).
The structure can be seen as a triangular lattice with a unit cell consisting of two atoms. a1 and a2
are the lattice vectors. The distance b between neighbor C atoms is 1.42Å. This determines the absolute value of lattice vectors a0 as 2.465 Å. Figure 5.1 (b) is the corresponding reciprocal lattice, and b1 and b2 are the lattice vectors. The first Brillouin zone is also a hexagon. The center of the first Brillouin zone is called Γ point. The corner points are denoted by K and named Dirac points. The middle points between Dirac points K are named M points.
Figure 5.1: Lattice structure of graphene.
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Figure 5.2: Left: Electronic dispersion of graphene. Right: enlarged view of the band structure close to one of the Dirac points. Reprinted figure with permission from Ref. (76). Copyright (2009) by the American Physical Society.
Electronic band structure. Figure 5.2 shows the band structure of graphene calculated from tight binding approximation (76). The upper and lower surface plots in the figure represent the conduction band and the valance band respectively. It can be seen in the figure that the conduction band and valence band meet at the Dirac points, leading to the zero band gap structure. The Fermi energy, also located at the Dirac points, is normally defined as the zero energy in the band structure. Near the Dirac points, the conduction band and the valence band are symmetric. Electrons whose energy is within about 1eV of the Dirac point have a linear dispersion relation. The dispersion relation at the Dirac points is expressed as:
𝐸±(𝒌) ≈ ±ℏ𝑣𝐹|𝒌 − 𝑲| (5.1)
k is the wave vector of the carrier. K is the wave vector for the Dirac point. 𝑣𝐹 ≈ 1 × 106 𝑚/𝑠 is the Fermi velocity. 𝐸±(𝒌) denote energies of the conduction band and valence band respectively.
The distinctive band structure is the primary reason for the unusual electronic properties of graphene. The linear dispersion relation shown in Eq. 5.1 resembles that of light. This means the charge carriers are analogues to relativistic particles and more easily and naturally described by the Dirac equation rather than by the Schrödinger equation. It provides a way to probe quantum electrodynamics phenomena by measuring graphene’s electronic properties (77). Graphene
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displays remarkable electron mobility at room temperature, leading to a resistivity of 10-6 Ω·cm, which is even lower than the resistivity of silver (78).
Figure 5.3: Phonon dispersion relation in graphite following surface cuts along lines Γ to M, M to K, and K to Γ in the first Brillouin zone. Black solid curves: results from DFT calculation within generalized gradient approximation (GGA). Black dash curves: results from DFT calculation within local density approximation (LDA). Yellow diamonds: data from neutron scattering. Green squares: data from electron energy loss spectrum (EELS). Blue squares: data from electron energy loss spectrum. Red squares: data from electron energy loss spectrum.
Black circles: data from X-ray scattering. Blue circles: data from double resonance Raman scattering. Red triangles: data from IR absorption. Reprinted from Ref. (79). Copyright (2004) with permission from Elsevier.
Phonon dispersion. Graphene is the second strongest material ever known. Its superlative status was recently replaced by another carbon allotrope carbine recently (80). Young’s modules as high as 1TPa have been measured on graphene (81). Graphene also has the highest thermal conductivity. These characteristics are all related to the unique phonon dispersion relation of graphene. Figure 5.3 shows theoretically calculated and experimentally measured phonon dispersion relation in graphite. It is expected to be very similar to graphene. ZA and ZO represent the out-of-plane acoustic and optical modes, respectively. LA and LO represent the in-plane longitudinal acoustic and optical mode. TA and TO are the in-in-plane sheer acoustic and optical modes. The lines are theoretical dispersion relations from DFT calculations. The dots
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represent experimental data obtained by various methods. The experimental measurements agree well with the theoretical calculation with the exception of the TA mode measurement along the Γ to M direction. This is because the crystal contains different rotational domains (79).
Figure 5.4: Illustration for the six force constant parameters. α1 and α2 are stretching force constants. γ1 and γ2 are bending force constant. δ is the twisting force constant and αs represents the interaction between the substrate and the graphene. Reprinted with permission from Ref. (73).
Copyright (1990) by the American Physical Society.
Besides ab initio calculations, a semi-empirical potential model proposed by T. Aizawa et al. is often used to calculate the phonon dispersion of graphene layers on different substrate (73). Six phenomenological force constants (FC) are used to describe graphene. As shown in Figure 5.4, α1 is the stretching FC between two nearest neighbor atoms. α2 is the stretching FC between the second nearest neighbors. γ1 represents a three body in-plane bending FC. γ2 represents a four body out-of-plane bending FC. δ is the twisting FC, which is similar with a force constant keeping the ethylene molecule flat. αs is a stretching FC between the graphene and the substrate.
The corresponding potential energy terms for the six force constants are:
𝛼
45 𝛾2
2 [(𝑢2𝑧+ 𝑢3𝑧+ 𝑢4𝑧− 3𝑢1𝑧)
|𝒓12| ]
2
(5.3) 𝛿
2[(𝑢5𝑧− 𝑢6𝑧) − (𝑢3𝑧− 𝑢4𝑧)
𝑎0 ]
2
(5.4) 𝛼𝑠
2 𝑢𝑧2 (5.5)
ui denotes the displacement vector of atom i. Vector rij indicates the relative mean position of atom i and atom j. a0 is the lattice constant. Eq. 5.1 is the stretching energy term for both α1 and α2. The lower case z in Eq. 5.2 and 5.3 indicates the vector component perpendicular to the graphene surface. Dispersion relation can be calculated based on these potential energy terms.
The six FCs are generated by fitting experimental data to the model.
Figure 5.5: Phonon dispersion for bulk graphite and graphene on TaC(111). The black dots are results from experimental measurements. The black curves are results from 6 FCs model calculation fitted to the data. Hollow squares in the left panel are data from neutron scattering.
In plane transverse modes are indicated by SH. The R branch in the right panel is due to the substrate. Reprinted with permission from Ref. (73). Copyright (2009) by the American Physical Society.
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